MinT - Best Known (228−80, 228, s)-Nets in Base 4

Best Known (228−80, 228, s)-Nets in Base 4

(228−80, 228, 163)-Net over F₄ — Constructive and digital

Digital (148, 228, 163)-net over F₄, using

1 times m-reduction [i] based on digital (148, 229, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 55, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (93, 174, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 87, 65)-net over F₁₆, using
      - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(228−80, 228, 240)-Net in Base 4 — Constructive

(148, 228, 240)-net in base 4, using

trace code for nets [i] based on (34, 114, 120)-net in base 16, using
- 1 times m-reduction [i] based on (34, 115, 120)-net in base 16, using
  - base change [i] based on digital (11, 92, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(228−80, 228, 514)-Net over F₄ — Digital

Digital (148, 228, 514)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(228−80, 228, 14171)-Net in Base 4 — Upper bound on s

There is no (148, 228, 14172)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 186472 709688 615199 991664 615741 594024 144537 375370 011767 164493 354284 451432 582095 905917 395060 254590 922841 504786 166403 469972 018531 115644 120342 > 4²²⁸ [i]